Chapter 3

Simplify the expression. \(\sin \left(\tan ^{-1} x\right)\)

Differentiate the function. $$ g(x)=\ln \frac{a-x}{a+x} $$

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. $$ f(x)=x^{2}-2 x $$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow \infty} \frac{\ln \sqrt{x}}{x^{2}}$$

Prove the identity. $$ \cosh x-\sinh x=e^{-x} $$

Simplify the expression. \(\cos \left(2 \tan ^{-1} x\right)\)

A sample of tritium-3 decayed to 94.5\(\%\) of its original amount after a year. (a) What is the half-life of tritium-3? (b) How long would it take the sample to decay to 20\(\%\) of its original amount?

Differentiate the function. $$ f(u)=\frac{u}{1+\ln u} $$

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. $$ f(x)=10-3 x $$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{t \rightarrow 1} \frac{t^{8}-1}{t^{5}-1}$$

Prove the identity. $$ \sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y $$

Scientists can determine the age of ancient objects by the method of radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon, \(^{14} \mathrm{C},\) with a half-life of about 5730 years. Vegetation absorbs carbon dioxide through the atmosphere and animal life assimilates \(^{14} \mathrm{C}\) through food chains. When a plant or animal dies, it stops replacing its carbon and the amount of 14 \(\mathrm{C}\) begins to decrease through radioactive decay. Therefore the level of radioactivity must also decay exponentially. A parchment fragment was discovered that had about 74\(\%\) as much \(^{14}\) C radioactivity as does plant material on the earth today. Estimate the age of the parchment.

Differentiate the function. $$ g(x)=\ln \left(x \sqrt{x^{2}-1}\right) $$

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. $$ g(x)=1 / x $$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{t \rightarrow 0} \frac{8^{t}-5^{t}}{t}$$

Prove the identity. $$ \cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y $$

A curve passes through the point \((0,5)\) and has the prop- erty that the slope of the curve at every point \(P\) is twice the \(y\) -coordinate of \(P .\) What is the equation of the curve?

Differentiate the function. $$ h(x)=\ln \left(x+\sqrt{x^{2}-1}\right) $$

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. $$ g(x)=\cos x $$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 0} \frac{e^{x}-1-x}{x^{2}}$$

$$\frac{d}{d x}\left(\cot ^{-1} x\right)=-\frac{1}{1+x^{2}}$$

Prove that \(\frac{d}{d x}\left(\cot ^{-1} x\right)=-\frac{1}{1+x^{2}}\)

Differentiate the function. $$ G(y)=\ln \frac{(2 y+1)^{5}}{\sqrt{y^{2}+1}} $$

Starting with the graph of \(y=e^{x},\) write the equation of the graph that results from (a) shifting 2 units downward (b) shifting 2 units to the right (c) reflecting about the \(x\) -axis (d) reflecting about the \(y\) -axis (e) reflecting about the \(x\) -axis and then about the \(y\) -axis

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{u \rightarrow \infty} \frac{e^{u / 10}}{u^{3}}$$

Prove the identity. $$ \frac{1+\tanh x}{1-\tanh x}=e^{2 x} $$

Prove that \(\frac{d}{d x}\left(\sec ^{-1} x\right)=\frac{1}{x \sqrt{x^{2}-1}}\)

In a murder investigation, the temperature of the corpse was \(32.5^{\circ} \mathrm{C}\) at \(1 : 30 \mathrm{PM}\) and \(30.3^{\circ} \mathrm{C}\) an hour later. Normal body temperature is \(37.0^{\circ} \mathrm{C}\) and the temperature of the surroundings was \(20.0^{\circ} \mathrm{C} .\) When did the murder take place?

Differentiate the function. $$ g(r)=r^{2} \ln (2 r+1) $$

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. $$ f(t) \text { is your height at age } t $$

Starting with the graph of \(y=e^{x},\) find the equation of the graph that results from (a) reflecting about the line \(y=4\) (b) reflecting about the line \(x=2\)

Differentiate the function. $$ F(s)=\ln \ln s $$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 0} \frac{x 3^{x}}{3^{x}-1}$$

Prove that \(\frac{d}{d x}\left(\csc ^{-1} x\right)=-\frac{1}{x \sqrt{x^{2}-1}}\)

Assume that \(f\) is a one-to-one function. (a) If \(f(6)=17,\) what is \(f^{-1}(17) ?\) (b) If \(f^{-1}(3)=2,\) what is \(f(2) ?\)

Find the domain of each function. $$f(x)=\frac{1-e^{x^{2}}}{1-e^{1-x^{2}}}$$ $$f(x)=\frac{1+x}{e^{\cos x}}$$

Differentiate the function. $$ y=\ln |\cos (\ln x)| $$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 0} \frac{\cos m x-\cos n x}{x^{2}}$$

Find the derivative of the function. Simplify where possible. $$y=\tan ^{-1}\left(x^{2}\right)$$

Find the domain of each function. $$g(t)=\sin \left(e^{-t}\right) \quad \text { (b) } g(t)=\sqrt{1-2^{t}}$$

Differentiate the function. $$ y=\tan [\ln (a x+b)] $$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 1} \frac{1-x+\ln x}{1+\cos \pi x}$$

If \(\cosh x=\frac{5}{3}\) and \(x>0,\) find the values of the other hyperbolic functions at \(x\).

Find the derivative of the function. Simplify where possible. \(y=\left(\tan ^{-1} x\right)^{2}\)

Differentiate the function. $$ H(z)=\ln \sqrt{\frac{a^{2}-z^{2}}{a^{2}+z^{2}}} $$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.$$\lim _{x \rightarrow 0} \frac{x}{\tan ^{-1}(4 x)}$$

Find the derivative of the function. Simplify where possible. $$g(x)=\sqrt{x^{2}-1} \sec ^{-1} x$$

(a) If \(\$ 1000\) is borrowed at 8\(\%\) interest, find the amounts due at the end of 3 years if the interest is compounded (i) annually, (ii) quarterly, (iii) monthly, (iv) weekly, (v) daily, (vi) hourly, and (vii) continuously. (b) Suppose \(\$ 1000\) is borrowed and the interest is compounded continuously. If \(A(t)\) is the amount due after \(t\) years, where \(0 \leqslant t \leqslant 3,\) graph \(A(t)\) for each of the interest rates \(6 \%, 8 \%,\) and 10\(\%\) on a common screen.

Differentiate the function. $$ f(x)=\left(x^{3}+2 x\right) e^{x} $$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 1} \frac{x^{a}-a x+a-1}{(x-1)^{2}}$$